Abstract

An out-tree $T$ is an oriented
tree with only one vertex of in-degree zero.
A vertex $x$ of $T$ is
internal if its out-degree is positive. We design randomized
and deterministic algorithms for deciding whether an input digraph
contains a given out-tree with $k$ vertices. The algorithms are of
runtime $O^*(5.704^k)$ and $O^*(5.704^{k(1+o(1))})$, respectively.
We apply the deterministic algorithm to obtain a deterministic
algorithm of runtime $O^*(c^k)$, where $c$ is a constant, for
deciding whether an input digraph contains a spanning out-tree with
at least $k$ internal vertices. This answers in affirmative a
question of Gutin, Razgon and Kim (Proc. AAIM'08).